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Flame zone and sub-surface reaction model for deflagrating RDX
FLAME Z O N E A N D S U B - S U R F A C E REACTION M O D E L FOR D E F L A G R A T I N G RDX M. BENREUVEN, L. H. CAVENY, R. J. VICHNEVETSKY,* AND M. SUMMERFIELD Princeton University, Princeton, New Jersey A study of 1,3,5 Trinitro Hexahydro 1,3,5 Triazine, RDX, burning as a monopropellant was undertaken to obtain a better understanding of the important chemical steps that control heat feedback to the condensed phase, to determine the contributions of the liquid layer, and to provide a means of evaluating theories for modifying the burning rate of nitramines. The following chemical mechanism is proposed: first, partial decomposition of the RDX molecule in the liquid phase; second, following vaporization, gas phase decomposition of RDX; third, oxidation of formaldehyde by NO 2. The flame structure and liquid layer reactions of deflagrating RDX were expressed in terms of the energy, continuity, and species equations corresponding to RDX decomposing in liquid and gaseous phases and the N O 2 / C H 2 0 reactions adjacent to the surface. In addition to the temperature profile and burning rate, the numerical solution provides the details of the interactions at the liquid/gas interface and the concentration profiles for the nine most prominent species. Using published kinetic data, the calculated results reveal that even though the liquid layer becomes thinner with increasing pressure, the increase in surface temperature causes its heat feedback contribution to increase. The pressure sensitivity of burning rate between 0,7 and 0.8 is interpreted in terms of the relative contributions of gas phase and liquid layer RDX decomposition and the oxidation of CH20. In particular, as pressure increases, the contribution from liquid layer reactions and the second order, N O e / C H 2 0 reaction become more prominent.
Introduction The increasing interest in nitramines (particularly 1,3,5 Trinitro Hexahydro 1,3,5 Triazine, RDX) as energetic p r o p e l l a n t components (in contrast to their usual role as secondary explosives) motivates this study. The goal of this study is to obtain an u n d e r s t a n d i n g of the chemical a n d physical mechanisms for the deflagration of RDX as a monopropellant. This is a necessary step toward d e v e l o p i n g insights into the potentially more complex problem of b u r n i n g RDX with p o l y m e r i c binders, as a composite propellant. More explicitly, the objectives for this study are: a. To define the important chemical steps and components of the mechanisms that control the heat feedback to the condensed phase.
*Consultant; Professor, Rutgers University 1223
b. To determine the contribution of the melt phase d e c o m p o s i t i o n process to the overall heat release d u r i n g deflagration. c. To investigate the distinct zones that occur in the gas phase flame at# low to moderate pressures. To achieve these goals, the physical a n d chemical elements that describe the deflagration events are c o m b i n e d into a flame model.
T h e Physical Model A steady-state, one-dimensional model for RDX deflagration is proposed, as illustrated in Fig. 1. As the constant velocity thermal wave passes through the solid, the m o n o p r o p e l l a n t is heated to its m e l t i n g point and develops a liquid layer in w h i c h p r i m a r y decomposition with its c o r r e s p o n d i n g heat release takes place. Then, following evaporation of the remaining liquid, RDX vapor and the gaseous products
1224
PROPELLANT IGNITION AND COMBUSTION
of primary decomposition transfer through a pre-mixed laminar deflagration zone; there, additional chemical reactions occur. The origin of the coordinate system, x = 0 is fixed at the liquid/gas boundary. In the gas phase, the region of interest extends from this boundary to a point where chemical processes cease to influence the heat feedback to the condensed phase, i.e., to the end of the so-called inner flame zone. Temperature, chemical species concentrations, and reaction rate are all variable through the liquid phase. In the gas, several simultaneous (overall) chemical reactions are considered; here, concentrations, temperature, reaction rates, as well as physical properties are all variable. One of the key elements of the present approach that differs from existing deflagration studies, is that the (variable) physical properties and chemical mechanism are based on available thermochemical data, without a priori adjustment by empirical or experimental burning rate data. This way, the results are not influenced by any presumptions about the very nature of the phenomena to be investigated. The benefits are that various chemical mechanisms and physical models for the processes involved may be examined for validity by comparisons with existing deflagration data. Further, those properties to which the model is most sensitive (and thus should be given with greater accuracy) may be identified readily.
Chemical Reactions A chemical reaction mechanism is proposed consisting of the following sequence (also illustrated in Fig. 1): a. Decomposition of RDX in the liquid phase. b. Following vaporization, gas-phase decomposition of RDX, the primary gasphase reaction. c. Simultaneously with (b.), oxidation of C H 2 0 by NOa is considered to be the secondary gas-phase reaction. Typically, the primary reactions yield N.2, N20, NO2, C H e O , and other products; the secondary reaction (c.) yields CO, CO2, H a O, and other products. The deflagration mechanism in this study has been formulated in terms of several overall chemical reactions. Two such overall reaction steps that yield product concentrations in good agreement with experimentally observed ones, are:
X,0)
I~i, RDX
TYPICAL~
I
CH2C
+
NO2
,LI.~--~ ~
OUTER FLAME ZONE,~ NO2
~'~'~I
/
I1"
~"
TEMPERAIIIKE--~
2
1 \\
I
CO CO2, H20, NO
II
\\
"+
~F
XF4 x"< XE
~ j / - INNER FLAMEZONE, (a)
SOLID
,xt
I
x
O+ ~. x ~ xF
RDX g a s e o u s flame model
TM AHv
-;m /~ REACTINGLAYER (RDX DECOMPOSITION) (b)
"I
\
GAS X
[NNERFLAMEZONE RDX~NO2, CH20, N20, NO CH20 + NO2 ~ , ,
Liquid layer and inner gaseous flame zone
F*c. 1. RDX flame model. 3 RDX--+ - - Nz + N 2 0 + NO2 + 3 C H 2 0 2 5 2 - - C H ~ O + NO2---* NO + - - C O 2 7 7 3 5 +--CO +--H20 7 7
(1)
(2)
Reaction (1) was assumed to hold for both gas and liquid phase decomposition of RDX. Although RDX decomposition occurs in both gas and liquid phases, the mechanism for the liquid phase reaction is probably different from the gas phase mechanism. Distinct liquid and gas RDX decomposition mechanisms have been proposed by Rauch and Fanelli, 1 Batten, ~ and Cosgrove and Owen. 3 In the gas, the initial step most likely involves N - - N bond scission; in the liquid, C - - N bond rupture might be the most likely initial step, z,a The unique mechanism proposed by Batten for the Iiquid 2 even suggests an intramolecular initial step, in which oxygen from one RDX molecule attacks (CH2:) from another, to form CHz O, NO~, N=O, plus non-volatile products (the presence of the latter, verified experimentally, could not be explained by any of the other models). A more complete discussion of the various chemical mechanisms may be found elsewhere. 4 Thermochemieal data for the overall rate of
REACTION MODEL FOR DEFLAGRATING RDX RDX pyrolytic d e c o m p o s i t i o n in the liquid phase was recorded b y several investigators (Refs. 1, 3, 6, 7, 8, 9, 10 a n d 11). The following Arrhenius-type rate constant was chosen for the liquid process: 8 klRDX (liquid) = 3 x 10 is exp(-47500/R,,T),
sec -~
(3)
Although direct RDX-vapor measurements were taken, 11 it seems more appropriate to e m p l o y the T N T (liquid) solution data 8 for the range of temperatures expected for the gas phase process. Therefore, klaox(gas)
=
10155
exp(-415OO/R,,T),
1225
of the secondary reaction occurs within the close-in flame zone. This introduces into the gaseous flame field the two-zone concept, d e p i c t e d in Fig. la: (i) an inner flame zone (or "near field") adjacent to the c o n d e n s e d surface, that contains r a p i d variations in all properties, and (2) an outer flame zone (or "far field") where properties are moderately varying in comparison, extending to a point where there are practically no more chemical or physical changes. The subject of the present study is the inner flame zone, where the strongest elements influencing the heat f e e d b a c k to the c o n d e n s e d phase are to be found. The following a posteriori observations are made:
sec 1 (4)
was used as a first estimate of an Arrhenius rate constant for gas-phase RDX decomposition. It should be noted that Rogers and D a u b 11 obtained m u c h lower activation energy a n d pre-exponential factor, in the range T = 500 to 526 K. Two sets of rate data were considered for the overall secondary Reaction (2). Pollard 12 estimated that above 430 K k 2 = 1012 e x p ( - 1 9 0 0 0 / R , T), cm 3/gmol-sec.
(5)
Filer13 used shock tube experiments to obtain the rate constant in a temperature range of 970 to i470 K: k 2 = 10 la'l e x p ( - 2 6 7 0 0 / R , T), cm 3 / g m o l - s e c
(6)
It can be shown that N 2 0 + C H e O reactions (due to higher activation energies) are sufficiently slow, so that their effect u p o n the near field in the gas (i.e., close to the g a s / l i q u i d surface) is negligible. Gas Phase Processes The consideration of several simultaneous chemical reactions in the gas phase raises the question of time scale (or associated length scale) uniformity. It was found that at least two such scales exist: r l - P~/l~v',t~ the characteristic time for the RDX decomposition _reaction in the vapor phase, a n d x2 - pf/Wry%i, that of a secondary reaction, e.g., C H 2 0 + NO2. It can be shown that -q < < ~2, with a difference of about two orders of magnitude. However, an important portion
a.
The "outer edge" of the inner flame zone (x = xs) was d e f i n e d as the p o i n t where the RDX d e c o m p o s i t i o n reaction is complete, i.e., t01f < 1 m o l / m 3 - s e c . Indeed, the gradients-of all properties at x = xf were found to be vanishingly small c o m p a r e d with those near the c o n d e n s e d surface. This further reinforces the concept that the processes that have the strongest influence on heat feedback to the c o n d e n s e d phase are i n c l u d e d in the thin inner flame zone, according to t h e foregoing definition of its outer b o u n d a ry.
b. The C H 2 0 + NO 2 reaction, although mostly slower and less energetic than the RDX reaction has a non-vanishing influence u p o n the heat feedback to the cond e n s e d phase, w h e n incorporated into the inner flame zone formulation. Therefore, the inner zone formulation m a y not be s i m p l i f i e d b y d r o p p i n g this ~econdary reaction without considerable sacrifice in accuracy. The reactive gas region of interest in the present deflagration m o d e l is considered essentially as a one-dimensional, premixed laminar flame. In setting up the formulation, the following assumptions were employed: pressure gradient and thermal gradient i n d u c e d diffusion, radiative transport and viscous effects were all considered negligible; all b i n a r y diffusion coefficients were taken equal to the diffusion coefficient of the mixture; the Lewis n u m b e r for i n d i v i d u a l species and for the mixture was taken as unity. The formulation consists of N-species a n d one energy conservation equation, the devel-
PROPELLANT IGNITION AND COMBUSTION
1226
opment of w h i c h m a y be followed from the basic formulation of Ref. 14. F o r O + -< x
d2 Y~ + - D dx 2 p
(pD)- m dx w, &
=d20 =dO + G--+ dx 2 dx
C
(v,k" 7 v,k)o~k P
k=l
For D
i= 1,2 .... N
d2T d 2( +--p p1 D [P~) -dcp -d+ - -xd x dx --1
M
Cpp
k=l
(7)
m]dTdx
N
(8)
'=
The overall mass continuity equation a n d the m o m e n t u m statement are, respectively: p V ~- m = const. P = const.,
(9) (10)
the latter b e i n g connnon in steady state deflagration studies. T h e equation of state for ideal gas mixture is N
P= pRu T E Y, / W~
(ll)
i=1
Equations (9)-(11) were also used to b r i n g the species and energy conservation equations to the form shown, Eqs. (7) and (8). Additionally, suitable approximations were introduced for C , = 1s C m I1/ a n d p D X/C,, in order to b r i n g about their explicit temperature d e p e n d e n c e : Cm - fi'i + / ~ i T
pD--
(T/rs) ~
(12) (13)
s
where a = 0.67 typically. The thermal conductivity h at the surface was evaluated u s i n g tabulated experimental thermochemieal data, 15 mainly according to procedures outlined in the literature. 16,17 A detailed discussion is provided in Ref. 4. Equations (7) a n d (8), after proper use of Eqs. (9) to (11) has been made, consist of N + I second order o r d i n a r y differential equations. The u n k n o w n variables, in vector form are:
Ur(x) = (Y1 Yz "" YN T)
The system is autonomous, coupled, a n d nonlinear. In the present study, the n u m b e r of chemical species considered is N = 9 and the n u m b e r of sinmltaneous (overall) chemical reactions is M = 2. The system of equations, written in tensor notation is:
(14)
_
F=0
(15)
where if7 is the tensor of diffusive term coefficients, G is the tensor of eonveetive term coefficients a n d 1~ is the vector of the p r o d u c tion terms. Generally, they all d e p e n d u p o n (7(x). The b o u n d a r y conditions are: [~(0) = [7~
(16a)
dO dx (xf) "- 0
(16b)
Thus, a 2-point b o u n d a r y value p r o b l e m has evolved for the gaseous flame zone, w h i c h m a y be solved for particular surface values, U s. Evidently, these values of 0"5 are not given in general and should also be found in the process of solution of the complete deflagration problem. Attention is called to the following points: a. The simplification, usually o b t a i n e d b y forming the suitable S h v a b - Z e l d o v i c h variables is not valid in general, due to the presence of several simultaneous chemical reactions. b. In terms of computational feasibility of the p r o p o s e d gas phase model, the analysis is not l i m i t e d b y the n u m b e r of simultaneous reactions as much as b y the total n u m b e r of species, N. c. There is clear advantage to solution in this manner, in contrast to models that c o m b i n e all chemical events into a single, overall reaction step. Namely, the present analysis reveals the relative effect of each reaction at any point of interest throughout the flame zone. Species and temperature profiles for converged solutions are depicted on Fig. 2; reaction rates r 1 a n d to 2 are plotted on a log scale in Fig. 3.
Numerical Solution for Gas Phase Equations A eonvenient means of obtaining a solution of the steady-state equations for the gas phase consists of f o r m u l a t i n g p s e u d o - n o n s t e a d y state' equations and solving them by a numerical time-marehing m e t h o d until steady-state is achieved.
REACTION MODEL FOR DEFLAGRATING RDX
In this expression, 2((U *) represents a matrix with coefficients that are functions of U* multiplying the solution vector U ~ The gen" eral implicit algorithm utilized to solve Eq. (18) is:
i
P = 40 AIM 108
1227
p = 20 ATM
(U *'§ - U * ' ) / A t = 0 ' X ( U ~
.~0D
~~o 104
J
~J'l
+ (1 - 0') 2 ( U ' J )
4 6 8 i0 DISTANCEFROMSURFACE,MICRONS
U*'---ZJ
(20)
12
Using a first predicted value of U *j+~
FIG. 2. Species concentrations and temperature profiles, inner flame zone, showing that increased RDX depletion near the surface together with larger gradients are observed as p increases. The conservation equation for species and energy may be written in vector form as:
O0/Ot =
(19)
[ I - O' at2(U*'+')] U *'+' =[I+(1-0')at2(U*')]
2 x,
U*'
Rearranging:
io2 0
U *'+,
L~ ((f)
+
F(0)
(17)
The method used for the numerical solution of these equations is based on approximation of the operators O/Ox and OZ/Ox2 in L 2 + F by central differences resulting in a system of ordinary differential equations:
d U * / d t = X(U*) U*
(18)
where U* is the solution vector expressed in a space of equidistant points across the flame.
U~ J+' = 2 U *~ - U *j-',
(21)
Equation (20) may be written in quasi-linear form as a corrector for U*J+~:
[I-
O' At)((U~J+')] U * ' '
= ZJ
(22)
and is then solved as a linear system by conventional block-tridiagonal Gaussian elimination. The right side of Eq. (22) was computed by the exact predictor: 18 Z j = Z j - 1 -.J- [ U Oj -
ZJ-l]//0
'
(23)
Initial conditions for this pseudo-nonsteady calculation include a temperature profile chosen to insure vigorous reactions and species mass fraction profiles that satisfy the relation ~--1 Y~ = 1 at any point and are physically reasonable.
Liquid Layer Analysis
6o0[ / ~ ~ .
-%~--~ ......
[/,:'.-"
to. ..........
~ 0
2 x,
to.~
7
~
/
......
4 6 8 10 DISTANCEFROMSURFACE,MICRONS
to., 0.1 12
Fro. 3. Reaction rate profiles, inner flame zone, showing increased relative importance of secondary reaction and more distinct inner-outer zone boundary at higher pressure.
Observations that confirm the existence of a liquid layer during steady burning of RDX at low to moderate pressures have been made by several investigators. 19,2~RDX decomposition in the liquid phase for temperatures above the melting point (480 K) has been verified experimentally. 6,s,n Scanning electron micrographs of extinguished samples el after burning at about 20 atm have been made showing entrapped gas bubbles; unfortunately, extinguishment was achieved by rapid depressurization, so one may not be certain about the actual presence of gas bubbles during the deflagration process itself. Detailed models of reacting melt layers have been developed for pyrolytic degradation (endothermic) of polymers z2 and incorporated into a deflagration model of a m m o n i u m perchlorate monopropellant, z3 The advantage
1228
PROPELLANT IGNITION AND COMBUSTION
to detailed reactive melt layer analysis is, in general, that Y1(0-), Y2(0-) ... YN(0-) and d T / d x ( O - ) are solved directly, without need to invoke additional simplifying assumptions (e.g., imposed burning rate data). Further, the analysis may apply to any small (but finite) liquid layer thickness. These may not be obtained from collapsed (zero thickness) reacting sub-surface approximate models that involve overall heat balance. In the remainder of this section, a brief description of the reacting liquid phase model is given. A full description of the development and solutions is given elsewhere. 4 RDX decomposition (to a significant extent) is considered to occur in the melt zone, with the corresponding heat release. This is the only reaction considered in the liquid. Its stoichiometry is given by Eq. (1) and the Arrhenius rate constant is that of Eq. (3). The gaseous products of this reaction are assumed to remain dissolved in the liquid (RDX) and to be convected by the liquid toward the (hot) fixed boundary at x = 0. Molecular diffusion (but not heat diffusion) is assumed to be negligible. The physical properties are assumed to be uniform throughout the liquid zone for the range of temperatures and extent of reaction considered. The conservation equations for the RDX species and energy are, in the region - ~ m+ ~< x_<0-:
dr,
r
A 1 e x p ( - E 1/ R u T )
(24)
dx d2T dx 2
r a
dT dx O~
= A 1e x p ( - E ~ / R , T ) Y ~ A h ~ , q - ~
x~
(25)
The boundary conditions are: r~(-aa) = 1
(26a)
T ( - 8 , . ) = T,.
(26b)
T(O) = T,
(26c)
The liquid layer thickness, 8 m, is not known a priori and has to be calculated. An overall energy balance on the solid phase plus the solid/liquid interface energy balance are utilized to provide additional conditions by which 8,~ is defined. These yield:
The method of solution, in general, involves transforming the given boundary value problem, Eqs. (24), (25), (26), and (27), into the associated initial value problem. Introducing dimensionless temperature and space coordinate: Xl =- ( T -
Ts)/T*
=- x r / e t c
(2Sa) (28b)
and the dimensionless parameters: T ~ ------Ahliq/WiCc, A 1 =---a c A l / r
01 =--- E 1 / B ~ T * , 02 =- T ~ / T * , 2 and a =- g m r / c % . The system
of equations becomes:
dlq dE
= - Y 1 A l e x p [ - 0 1 / ( - q +02)] d~l --= dE
dv
--= dE
v(E)
v(E) + Y, A1 exp[-O~/('q +Oz)]
(29a) (29b) (29c)
The initial conditions are: Y1 ( - a ) = 1 rl(-a) = (T m - Ts)/T* v(-a)
= [Ah m + C c ( T ~ - T o ) ] ~ C o T *
(30a) (SOb) (30c)
The system of Eqs. (29) and (30) may be integrated numerically with little effort. Note that although only YI(E) is calculated, the r e m a i n i n g species concentrations may be found at any point using the stoichiometry of Eq. (1). "q(~), being a continuous, monotonous, single-valued function of ~ in the region - a -< E -< 0-, serves as a measure of the point where the integration is stopped; viz., where Xl = 0(or T = Ts). To obtain a qualitative measure of the solution behavior, a more elementary model was set up. By assuming all heat release to be localized at the x = 0 interface, an analytical solution was obtained using asymptotic methods. The elementary model clearly approximates cases where only a small amount of distributed RDX decomposition occurs in the melt, i.e., a "thin" melt zone. Results of the comprehensive model compared with the elementary model are shown in Figs. 4, 5, and 6.
dT Gas~Melt Interface Conditions
x~-~x ( - ~ ) = m[Ah m + Cc(T m-
To)]
(27)
In the foregoing sections, the gas phase and melt phase processes were treated separately.
REACTION MODEL FOR DEFLAGRATING RDX 465
i
COMPREHENSIVEMODEL APPROXIMATEANALYSIS . . . . . . . . . BOTH CONVERGED(T~,M') SOLUTIONS--(~--
~-,20,0
390
"
i
MASSFLUX
i
i
PRESSURE
?
E
~
1229
315
15.0
0,52 I - - I
z
\\
/ ~ ......
240
\~
165 rMAS~ FLUX PRESSURE I G/CM'LSEC
O, fl / 600
ATM
I I L I 620 640 660 680 Ts, SURFACETEMPERATURE(K)
Fro. 4. Liquid phase layer thickness vs temperature showing decelerated decrease in thickness as regression rate becomes larger. So far, the only parameters they seem to have in c o m m o n are T~ a n d m (for any particular case). Evidently, there is a stronger int e r d e p e n d e n c e b e t w e e n those two elements, in terms of reactant s u p p l y to the gas b y the liquid phase and heat f e e d b a c k from the gas to the liquid. The objective of this section is to describe the l i q u i d / g a s interface conservation conditions and the method e m p l o y e d to incorporate them into the overall deflagration model. The following energy and species conservation conditions s h o u l d be satisfied at the liqu i d / g a s interface, x = 0:
MASSFLUX PRESSURE G/CN2-SEC ATM
" ' - , , . - " - , , , ~ . ~z'~,,,.(40,
.....
921
700
S. . . . . .
620 640 660 680 Ts, SURFACETEMPERATURE(K)
700
FIG. 6. Liquid phase thermal gradients vs temperature showing accelerated increase in dT/dx (0-) as regression rate increases.
dT
dT
% ~-(o+)= xo ~-(o-) + m , ~ h ~ Y, ( 0 -
Y, (0 + )
) (31)
(oD) s dY, - (0 + ) = Y, ( 0 - ) ,
m
dx
i = 1, 2 . . . . N
(32)
In addition, a C l a u s i u s - C l a p e y r o n evaporation law is utilized:
Pvl = Pv~ e x p
- -~u
T~
T O.
(33)
T O and P~ 1 are the reference temperature and the c o r r e s p o n d i n g reference partial pressure of RDX, taken at the melting point. Equations (31) to (33) constitute s system of N + 2 algebraic constraints, involving the following unknowns: Y~(0-),
Y~(0+),
(pD)~dY~/dx(O+), for
i= 1,2,...N
and 0,7
Xg~ dT/dx(O+), '~
0,6%
do
640
d0
dT/dx(O-), T~ and
m
(10)\"x~
680
Ts, SURFACETEMPERATURE(K)
FIG. 5. Liquid phase RDX mass friction vs temperature showing that higher T, of converged solutions overbalances smaller residence time to obtain larger RDX depletion at higher regression rates.
The formulation contains 3 N + 4 unknowns. This n u m b e r is r e d u c e d b y the following considerations: a. Given T~ and m, the solution of the l i q u i d
phase provides Yi(O-), i = 1, 2 . . . . and d r / dx(O- ).
N
PROPELLANT IGNITION AND COMBUSTION
1230
b. Given T,, m and Y~(0+), i = 1, 2 . . . . N [plus the boundary conditions (Eq. 16.b) at x = + % naturally] the solution of the gas phase yields k g, d T / d x (0 + ) and (oD)~ dY, / dx (0 + ), i = 1, 2 . . . . N. Therefore, a system of N + 2 equations with N + 2 unknowns remains, thus resolving the problem of closure. The unknowns are considered in the present framework as the set of independent parameters. Explicitly, their vector is defined as: 12r= (Y~(0+)... YN(0 +) T , m )
(34)
Equations (31)-(33) may now be written in short notation as the system: fj(~)=0,
j = 1,2 .... N + 2
(35)
Note that (in the nontrivial case) fi(0-*) = 0 for all j, only if all the interface conditions are satisfied, i.e., the value of ~* corresponds to a steady-state solution of the deflagration problem at the given ambient conditions. This, of course, is based upon the notion that such a solution is unique, if its exists. In practice, the following iterative procedure is suggested in order to arrive at the desired solution vector, ft*. The process begins by initially approximating ~0, for which gas and liquid phase solutions are generated; then, an attempt to satisfy Eq. (35) is made, resulting generally in: fj(~0)__ejo#0 '
j=l,
2,...N+2
(36)
The residuals or errors ~ 0 indicate that 120 ~ ~*, i.e., that 12~ is n ~ the ~ solution. The next trial vector ~ is obtained by using the Newton-Raphson iterative method; the process is repeated until sufficient convergence, (i.e., g --~ 0) is achieved. To simplify this rather complex general case for practical computations, only 3 out of the total of N + 2 = 11 constraints were actively iterated upon; these were chosen as the most sensitive constraints. Correspondingly, the dimension of ~ was also reduced to 3, taking the most deeply e m b e d d e d parameters; the rest were made dependent upon these 3. The 3 conditions chosen were the energy conservation, RDX species conservation and the evaporation law for RDX. (The remaining constraints were only observed.) The 3 independent parameters chosen were 1:1(0 + ), T~ and m. The Jacobian matrix for the iterative procedure is now defined
J -
(37)
It is formed numerically by repeatedly perturbing the vector 0- by 8~/k) and obtaining the corresponding 8~ Ik), then used in finite difference approximations of 0k j / 0 ~ . Corrections to the (n + 1) iteration-loop Ix are given by 8~.<"+l> = - J - ~ ~ <">
(38)
where (n) denotes the current iteration loop. The t2rocess is repeated until convergence, i.e., --* 0 is obtained. The procedure, being convergent generally, was found to diverge at the vicinity of minimal ]gl. This is due to ~ not being a single-valued function of ~ (although ~ is single valued in ~,). The transformation of Eq. (38) in the vicinity of minimal I~1, is thus not unique without J being a singular transform. In order to overcome this difficulty, the process of finding/2* at minimal 1~1was aided by marching in ~ in order to obtain the corresponding ~. In the region of minimal j~], e 3 that corresponds to the evaporation law, was at least 2 orders of magnitude smaller than % and %. Thus, marching by T, and m (Y1 (0 +) being generated by the evaporation law) proved sufficient for obtaining a map of {(g). Converged solutions are therefore defined by [e,(r**; m*)l = I%(Ts*; m*)l = minimum. Discussion of Results
Converged steady-state solutions tend to have mass fluxes reasonably close to those observed experimentally at the pressures considered. As expected, the margin of uncertainty in both T, and m, as well as the minimal ]El attainable, increase with pressure. Maximal components of 4 are always those corresponding to energy and RDX conservation constraints, Eqs. (31) and (32); at the points of convergence, these amount to 0.04, 0.10, and 0.16 at p = 10, 20, and 40 atm, respectively. Gas phase solutions at p = 10, 20, and 40 atm are plotted in Figs. 2 and 3. For purposes of demonstration the corresponding experimentally observed m were used; these values of m are always within the region of minimal I~l in ~-space, found in this study. Figure 2 shows mass fractions and temperature profiles in the inner flame zone. NO, CO 2, CO and H 2 0 were omitted here because of their small concentrations. It may be observed that gradients become steeper and xf becomes smaller
REACTION MODEL FOR DEFLAGRATING RDX as p increases. This is even more evident in Fig. 3, where o)1 and o)2 profiles are plotted on a log scale. Figures 2 and 3 clearly show that the division of the gaseous flame field into 2 zones becomes more distinct as p increases. To verify the importance of including the secondary reaction, an artificial suppression (by a factor of 0.01) of the C H z O + NO 2 reaction rate resulted in reduced heat feedback to the condensed phase by 7%, 11%, and 16% at p = 10, 20, and 40 atm, respectively, which confirms that the secondary reaction is important in the inner flame zone. Liquid phase solutions are plotted in Figs. 4, 5, and 6. In addition, lines connecting converged (T s, m) solutions were overplotted. Figure 4 shows that the layer thickness and residence time (pc?3m/rrt) decrease with increasing p. Figure 5 indicates that the fraction of RDX reacted in the liquid increases with increasing p; this is a result of two opposing trends: the shorter residence time at higher p decreases RDX decomposition, while higher T~ at higher p increases RDX decomposition. Thus it is seen that the higher T~ corresponding to large p overbalances in favor of more RDX decomposition at the higher pressures. Finally, Fig. 6 shows that XcdTfdx increases in an accelerated manner as p increases.
lated, these can be tested readily within the framework of the present model.
Nomenclature As Ai,/~ C D E~ F f, -~h i Ahti q Ah .... I J k
L 2
M
Conclusions m
A model for pure RDX deflagration was formulated and tested. It describes the processes involved in a reacting melt layer and the inner flame zone, consisting of two simultaneous reactions. A set of calculations were carried out using published kinetic and thermochemical data and Arrhenius-type chemical rates. The trends are in good agreement with the published deflagration data. The plausibility of distinct flame zones at intermediate pressures was demonstrated by solving the comprehensive mathematical model. For completeness, the outer flame zone processes should also be formulated and the solution incorporated by matching the outer and inner flame regions at the x = xf interface. In view of the inner flame zone model solution, this may be done with relative ease. Clearly, the model is most sensitive to the thermochemical and kinetic data, which are not known with great certainity and are not expected to be accurate over the full range of pressures and temperatures. However, as more complete data are obtained and more sophisticated chemical mechanisms are postu-
1231
N P R~ r T U U* V v W~, W x .~ y~
pre-exponential factor; for first order reaction, s e e - ' ; for second order reaction, cm3/grnol-sec coefficients in empirical (linear) equation for specific heat, Co; cal/g-K and cal/g-K 2, respectively heat capacity, cal/g-K diffusion coefficient, c m 2 / s e c activation energy, c a l / m o l source term vector in numerical scheme liquid/gas interface constraint specific heat of formation, of species i, c a l / g specific heat of reaction in liquid, cal/g specific heats of fusion, of vaporization respectively, c a l / g the identity matrix Jacobian matrix, Eq. (37) reaction rate constant; for first order reaction, sec-1; for second order, cm 3/ gmol-sec second order differential operator, xspace total number of simultaneous reactions mass burning rate or mass flux, g /
cm2-sec total number of species pressure, atm. (Note: in equation of state--in c.g.s, units.) universal gas constant, 1.986 cal/ gmo]-K. (Note: in equation of state--in equivalent c.g.s, units). linear regression rate, cm/sec temperature, K solution vector, gas phase. numerical (finite difference) approximation of the solution vector U velocity of gas, c m / s e c dimensionless temperature gradient, Eq. (29b) molecular weight of species i and average molecular weight of mixture respectively, g/tool distance, cm finite difference space operator mass fraction
Greek S~tmbols ct
thermal diffusivity, c m 2 / s e c
1232 ~rn
At e, Ej
PROPELLANT IGNITION AND COMBUSTION melt layer thickness, cm time i n c r e m e n t in quasi-nonsteady numerical scheme vector of residuals and c o m p o n e n t thereof, respectively, in g a s / l i q u i d interface constraints, Eq. (36) rx/e~r d i m e n s i o n l e s s space coordinate in l i q u i d layer (T Ts)/(Ahli q / Cc) , d i m e n s i o n l e s s t e m p e r a t u r e in l i q u i d layer implicit parameter in quasi-nonsteady numerical scheme, 0 -< 0' -< - -
0'
1
k P., P.j
P 1) ik ~k
thermal conductivity, c a l / c m - s e c - K vector of i n d e p e n d e n t parameters, Eq. (34) and c o m p o n e n t thereof, respectively. density, g / c m a stoiehiornetric coefficient of i th species in the k th reaction k th overall reaction rate, t o o l / c m 3 - s e c
Subscripts c
e,f g i j
k p s 0 1
c o n d e n s e d phase denote ends of outer and inner flame zones, respectively gas phase i th species jth conservation constraint at g a s / l i q u i d interface, Eq. (35) k th overall reaction constant pressure g a s / l i q u i d surface a m b i e n t conditions; also, p r e d i c t e d solutions in numerical scheme denote RDX species, i.e., i = 1
"2. BATTEN,T. T.: Australian J. Chem. 23, 737 (1970). Also, Aus. J. Chem. 23, 749 (1970) and Dept. of Supply, Australian Defense Scientific Service, Defense Standards Laboratories, Rept. No. 412 (1971). 3. CoscRovE, J. D. AND OWEN, A. J.: Combustion and Flame, 22, 13 (1974) and also Combustion and Flame, 22, 19 (1974). 4. BENREUVEN,M.: Ph.D. Dissertation in preparation, Princeton University, Princeton, N.J. 5. FLAXAGA.~,D. A,: Personal communication, January 1976. 6. HALL,P. G.: Trans. Farad. Soc., 67 (Pt. 2), 556 (1971). 7. ROSEN, J. M, AND DICKINSON,D.: J. Chem. Eng. Data, 14, 120 (1969). 8. ROBERTSON,A. J. B., Trans. Farad. Sot,, 45, 85 (1949). 9. SMITH,B. S.: "DTA and DTGA Analysis Studies of RDX and HMX," NWL Technical Report, TR-2316, July 1969, U.S. Naval Weapons Laboratory, Dahlgren, VA. 10. ROGERS,R. N. ANDSMITH,L. C.: Thermochemica Acta, 1970, pp. 1-9. 11. ROGERS, R. N. AXD DArn, G. W.: Anal. Chem. 45, 3, 1973, p. 596. 12, POLLARD,F. H. AND WYATT, R. M. H.: Trans. Farad. Soc., 45, 760 (1949). 13. FWER, R.: Personal Communication, PittmanDun Laboratory, Frankford Arsenal, Philadelphia, Pa., Nov. 1075. 14. WILLIAMS, F. A.: Combustion Theortj--The
Fundamental Theo~ of Chemically Reacting Flow Systems, pp. 1-17, 390-403, Addison15.
Superscripts 0
(k) (n) J T
initial a p p r o x i m a t i o n in g a s / m e h interface constraints, iteration scheme p e r t u r b a t i o n n u m b e r in iteration scheme iteration number, Eq. (38) n u m b e r of time step in n u m e r i c a l scheme, t = j A t transpose (for vector or matrix)
16. 17. 18.
19. REFERENCES 1. RAUCH,F. C. AND FANELLI,A. J.: J. Phys. Chem, 73, 1604 (1969).
20.
21.
Wesley Publishing Co., Inc., Reading, Mass., 1965. TOULOUKIAN,Y. S., LILLY, P. E. AND SAXENA,S. C.: Thermophysical Properties of Matter, the TPRC Data Series, IFI/Plenum, N.Y.-Wash., 1970. Also: Gallant, R. W., Physical Properties of Hydrocarbons, C1-C4, Aldehydes, Gulf Publishing Co., Houston, Texas, 1968. BIRD, R. B., STEWART,W. E. AND L1GHTFOOT,E. N.: Transport Phenomena, Chapt. 8, pp. 243264, J. Wiley, N.Y., 1960. SHELVA,R. A., McBmDE, B. J.: NASA TND-7056, NASA, Washington, D.C., Jan. 1973. VICHNEVETSk'Y,R.: Report NAM 153, Department of Computer Science, Rutgers University, New Brunswick, N.J., August 1974. TAYLOR,J.: Trans. Faraday Soc., 58, 471, pp. 561-568, 1962. BOBOLEV,V. K., MARGOLIN,A. M. AND CHUIKO, S. V.: ]. of Combustion, Explosion and Shock Waves, Vol. 2, No. 4, pp. 24-32, 1964. DERB,R. L., BOGGS,T. L., ZVRN,D. E. ANDDIBBLE, E. J.: Proceedings of the 11th JANNAF Combus-
REACTION M O D E L FOR D E F L A G R A T I N G RDX
tion Meeting, Vol. 1, pp, 231-242, CPIA Publication #261, Chemical Propulsion Information Agency, Sept. 1974.
1233
22. LAYCEt.LE, G.: AIAA J., 8, 1989-1996, (1970). 23. G~-lP,nO, C. AXD WILLIAMS, F. A.: AIAA J., 9, 1345-1356 (1971).
COMMENTS N. Kubota, Japan Defense Agency, Japan. The deflagration mechanism of RDX was described in terms of two reaction steps expressed by Eqs. (3) and (4), and the reactions between NaO and C H 2 0 were assumed to be sufficiently slow to affect the rate of the deflagration in this paper, However, the secondary gas phase reaction involving NO reaction is expected to be strong exothennic reaction, and the heat feedback from this secondary reaction zone to the first reaction zone possibly accelerates the reaction expressed by Eqs. (3) and (4). Did you examine the effect of this secondary reaction on the deflagration rate of RDX? The gas phase reactions involving NO is considered to be fast enough when the temperature and the pressure are high. The photographic observations of RDX crystal show that the luminous flame of RDX appears very close to the b u r n i n g surface of the crystal and is not like the case of the luminous flames of nitrate ester based propellants. Authors'Reply. The present paper's main concern, in terms of gas phase analysis is the "near-field" flame zone, where gaseous RDX decomposition and NO 2 + C H 2 0 reactions dominate. "Outer" flame field processes, in comparison, may have higher order, (namely smaller) effects upon the overall deflagration rate and the heat feedback to the condensed phase.
In particular, NO + C H 2 0 reaction is quite improbable due to high expected activaton energy, even at temperatures in the neighborhood of 1000~ (Notethat the N~----Ob o n d energy is 149.7 kcal/mole, being quite high at the pressure range considered.) To demonstrate this point with a reaction that is more probable at the outer field, consider the reaction rate of N~O + C H e O at the end of the inner flame zone; using the converged results of a 20 atm run and available data, approximately: A 3 - 1.2 • 1013 c m a / g m o l - s e c
and
E 3 - 44000 c a l / g m o l SO: toz = 72 x ka(1200 ) x (0.21/0.044)(0.42/0.030) = 3.8 x 102 mol/ma-sec while for NO 2 + C H 2 0 at the same point, ~0z 10 v mol/ma-sec. Therefore, the heat produced b y the N 2 0 + C H 2 0 reaction may not affect the thermal gradient (at the point considered) appreciably--at least not to first order. Finally, it should be noted that the t h i n luminous zone adjacent to the condensed surface is not observed for pure nitramine crystals deflagrating at 21 atm.
Introduction The increasing interest in nitramines (particularly 1,3,5 Trinitro Hexahydro 1,3,5 Triazine, RDX) as energetic p r o p e l l a n t components (in contrast to their usual role as secondary explosives) motivates this study. The goal of this study is to obtain an u n d e r s t a n d i n g of the chemical a n d physical mechanisms for the deflagration of RDX as a monopropellant. This is a necessary step toward d e v e l o p i n g insights into the potentially more complex problem of b u r n i n g RDX with p o l y m e r i c binders, as a composite propellant. More explicitly, the objectives for this study are: a. To define the important chemical steps and components of the mechanisms that control the heat feedback to the condensed phase.
*Consultant; Professor, Rutgers University 1223
b. To determine the contribution of the melt phase d e c o m p o s i t i o n process to the overall heat release d u r i n g deflagration. c. To investigate the distinct zones that occur in the gas phase flame at# low to moderate pressures. To achieve these goals, the physical a n d chemical elements that describe the deflagration events are c o m b i n e d into a flame model.
T h e Physical Model A steady-state, one-dimensional model for RDX deflagration is proposed, as illustrated in Fig. 1. As the constant velocity thermal wave passes through the solid, the m o n o p r o p e l l a n t is heated to its m e l t i n g point and develops a liquid layer in w h i c h p r i m a r y decomposition with its c o r r e s p o n d i n g heat release takes place. Then, following evaporation of the remaining liquid, RDX vapor and the gaseous products
1224
PROPELLANT IGNITION AND COMBUSTION
of primary decomposition transfer through a pre-mixed laminar deflagration zone; there, additional chemical reactions occur. The origin of the coordinate system, x = 0 is fixed at the liquid/gas boundary. In the gas phase, the region of interest extends from this boundary to a point where chemical processes cease to influence the heat feedback to the condensed phase, i.e., to the end of the so-called inner flame zone. Temperature, chemical species concentrations, and reaction rate are all variable through the liquid phase. In the gas, several simultaneous (overall) chemical reactions are considered; here, concentrations, temperature, reaction rates, as well as physical properties are all variable. One of the key elements of the present approach that differs from existing deflagration studies, is that the (variable) physical properties and chemical mechanism are based on available thermochemical data, without a priori adjustment by empirical or experimental burning rate data. This way, the results are not influenced by any presumptions about the very nature of the phenomena to be investigated. The benefits are that various chemical mechanisms and physical models for the processes involved may be examined for validity by comparisons with existing deflagration data. Further, those properties to which the model is most sensitive (and thus should be given with greater accuracy) may be identified readily.
Chemical Reactions A chemical reaction mechanism is proposed consisting of the following sequence (also illustrated in Fig. 1): a. Decomposition of RDX in the liquid phase. b. Following vaporization, gas-phase decomposition of RDX, the primary gasphase reaction. c. Simultaneously with (b.), oxidation of C H 2 0 by NOa is considered to be the secondary gas-phase reaction. Typically, the primary reactions yield N.2, N20, NO2, C H e O , and other products; the secondary reaction (c.) yields CO, CO2, H a O, and other products. The deflagration mechanism in this study has been formulated in terms of several overall chemical reactions. Two such overall reaction steps that yield product concentrations in good agreement with experimentally observed ones, are:
X,0)
I~i, RDX
TYPICAL~
I
CH2C
+
NO2
,LI.~--~ ~
OUTER FLAME ZONE,~ NO2
~'~'~I
/
I1"
~"
TEMPERAIIIKE--~
2
1 \\
I
CO CO2, H20, NO
II
\\
"+
~F
XF4 x"< XE
~ j / - INNER FLAMEZONE, (a)
SOLID
,xt
I
x
O+ ~. x ~ xF
RDX g a s e o u s flame model
TM AHv
-;m /~ REACTINGLAYER (RDX DECOMPOSITION) (b)
"I
\
GAS X
[NNERFLAMEZONE RDX~NO2, CH20, N20, NO CH20 + NO2 ~ , ,
Liquid layer and inner gaseous flame zone
F*c. 1. RDX flame model. 3 RDX--+ - - Nz + N 2 0 + NO2 + 3 C H 2 0 2 5 2 - - C H ~ O + NO2---* NO + - - C O 2 7 7 3 5 +--CO +--H20 7 7
(1)
(2)
Reaction (1) was assumed to hold for both gas and liquid phase decomposition of RDX. Although RDX decomposition occurs in both gas and liquid phases, the mechanism for the liquid phase reaction is probably different from the gas phase mechanism. Distinct liquid and gas RDX decomposition mechanisms have been proposed by Rauch and Fanelli, 1 Batten, ~ and Cosgrove and Owen. 3 In the gas, the initial step most likely involves N - - N bond scission; in the liquid, C - - N bond rupture might be the most likely initial step, z,a The unique mechanism proposed by Batten for the Iiquid 2 even suggests an intramolecular initial step, in which oxygen from one RDX molecule attacks (CH2:) from another, to form CHz O, NO~, N=O, plus non-volatile products (the presence of the latter, verified experimentally, could not be explained by any of the other models). A more complete discussion of the various chemical mechanisms may be found elsewhere. 4 Thermochemieal data for the overall rate of
REACTION MODEL FOR DEFLAGRATING RDX RDX pyrolytic d e c o m p o s i t i o n in the liquid phase was recorded b y several investigators (Refs. 1, 3, 6, 7, 8, 9, 10 a n d 11). The following Arrhenius-type rate constant was chosen for the liquid process: 8 klRDX (liquid) = 3 x 10 is exp(-47500/R,,T),
sec -~
(3)
Although direct RDX-vapor measurements were taken, 11 it seems more appropriate to e m p l o y the T N T (liquid) solution data 8 for the range of temperatures expected for the gas phase process. Therefore, klaox(gas)
=
10155
exp(-415OO/R,,T),
1225
of the secondary reaction occurs within the close-in flame zone. This introduces into the gaseous flame field the two-zone concept, d e p i c t e d in Fig. la: (i) an inner flame zone (or "near field") adjacent to the c o n d e n s e d surface, that contains r a p i d variations in all properties, and (2) an outer flame zone (or "far field") where properties are moderately varying in comparison, extending to a point where there are practically no more chemical or physical changes. The subject of the present study is the inner flame zone, where the strongest elements influencing the heat f e e d b a c k to the c o n d e n s e d phase are to be found. The following a posteriori observations are made:
sec 1 (4)
was used as a first estimate of an Arrhenius rate constant for gas-phase RDX decomposition. It should be noted that Rogers and D a u b 11 obtained m u c h lower activation energy a n d pre-exponential factor, in the range T = 500 to 526 K. Two sets of rate data were considered for the overall secondary Reaction (2). Pollard 12 estimated that above 430 K k 2 = 1012 e x p ( - 1 9 0 0 0 / R , T), cm 3/gmol-sec.
(5)
Filer13 used shock tube experiments to obtain the rate constant in a temperature range of 970 to i470 K: k 2 = 10 la'l e x p ( - 2 6 7 0 0 / R , T), cm 3 / g m o l - s e c
(6)
It can be shown that N 2 0 + C H e O reactions (due to higher activation energies) are sufficiently slow, so that their effect u p o n the near field in the gas (i.e., close to the g a s / l i q u i d surface) is negligible. Gas Phase Processes The consideration of several simultaneous chemical reactions in the gas phase raises the question of time scale (or associated length scale) uniformity. It was found that at least two such scales exist: r l - P~/l~v',t~ the characteristic time for the RDX decomposition _reaction in the vapor phase, a n d x2 - pf/Wry%i, that of a secondary reaction, e.g., C H 2 0 + NO2. It can be shown that -q < < ~2, with a difference of about two orders of magnitude. However, an important portion
a.
The "outer edge" of the inner flame zone (x = xs) was d e f i n e d as the p o i n t where the RDX d e c o m p o s i t i o n reaction is complete, i.e., t01f < 1 m o l / m 3 - s e c . Indeed, the gradients-of all properties at x = xf were found to be vanishingly small c o m p a r e d with those near the c o n d e n s e d surface. This further reinforces the concept that the processes that have the strongest influence on heat feedback to the c o n d e n s e d phase are i n c l u d e d in the thin inner flame zone, according to t h e foregoing definition of its outer b o u n d a ry.
b. The C H 2 0 + NO 2 reaction, although mostly slower and less energetic than the RDX reaction has a non-vanishing influence u p o n the heat feedback to the cond e n s e d phase, w h e n incorporated into the inner flame zone formulation. Therefore, the inner zone formulation m a y not be s i m p l i f i e d b y d r o p p i n g this ~econdary reaction without considerable sacrifice in accuracy. The reactive gas region of interest in the present deflagration m o d e l is considered essentially as a one-dimensional, premixed laminar flame. In setting up the formulation, the following assumptions were employed: pressure gradient and thermal gradient i n d u c e d diffusion, radiative transport and viscous effects were all considered negligible; all b i n a r y diffusion coefficients were taken equal to the diffusion coefficient of the mixture; the Lewis n u m b e r for i n d i v i d u a l species and for the mixture was taken as unity. The formulation consists of N-species a n d one energy conservation equation, the devel-
PROPELLANT IGNITION AND COMBUSTION
1226
opment of w h i c h m a y be followed from the basic formulation of Ref. 14. F o r O + -< x
d2 Y~ + - D dx 2 p
(pD)- m dx w, &
=d20 =dO + G--+ dx 2 dx
C
(v,k" 7 v,k)o~k P
k=l
For D
i= 1,2 .... N
d2T d 2( +--p p1 D [P~) -dcp -d+ - -xd x dx --1
M
Cpp
k=l
(7)
m]dTdx
N
(8)
'=
The overall mass continuity equation a n d the m o m e n t u m statement are, respectively: p V ~- m = const. P = const.,
(9) (10)
the latter b e i n g connnon in steady state deflagration studies. T h e equation of state for ideal gas mixture is N
P= pRu T E Y, / W~
(ll)
i=1
Equations (9)-(11) were also used to b r i n g the species and energy conservation equations to the form shown, Eqs. (7) and (8). Additionally, suitable approximations were introduced for C , = 1s C m I1/ a n d p D X/C,, in order to b r i n g about their explicit temperature d e p e n d e n c e : Cm - fi'i + / ~ i T
pD--
(T/rs) ~
(12) (13)
s
where a = 0.67 typically. The thermal conductivity h at the surface was evaluated u s i n g tabulated experimental thermochemieal data, 15 mainly according to procedures outlined in the literature. 16,17 A detailed discussion is provided in Ref. 4. Equations (7) a n d (8), after proper use of Eqs. (9) to (11) has been made, consist of N + I second order o r d i n a r y differential equations. The u n k n o w n variables, in vector form are:
Ur(x) = (Y1 Yz "" YN T)
The system is autonomous, coupled, a n d nonlinear. In the present study, the n u m b e r of chemical species considered is N = 9 and the n u m b e r of sinmltaneous (overall) chemical reactions is M = 2. The system of equations, written in tensor notation is:
(14)
_
F=0
(15)
where if7 is the tensor of diffusive term coefficients, G is the tensor of eonveetive term coefficients a n d 1~ is the vector of the p r o d u c tion terms. Generally, they all d e p e n d u p o n (7(x). The b o u n d a r y conditions are: [~(0) = [7~
(16a)
dO dx (xf) "- 0
(16b)
Thus, a 2-point b o u n d a r y value p r o b l e m has evolved for the gaseous flame zone, w h i c h m a y be solved for particular surface values, U s. Evidently, these values of 0"5 are not given in general and should also be found in the process of solution of the complete deflagration problem. Attention is called to the following points: a. The simplification, usually o b t a i n e d b y forming the suitable S h v a b - Z e l d o v i c h variables is not valid in general, due to the presence of several simultaneous chemical reactions. b. In terms of computational feasibility of the p r o p o s e d gas phase model, the analysis is not l i m i t e d b y the n u m b e r of simultaneous reactions as much as b y the total n u m b e r of species, N. c. There is clear advantage to solution in this manner, in contrast to models that c o m b i n e all chemical events into a single, overall reaction step. Namely, the present analysis reveals the relative effect of each reaction at any point of interest throughout the flame zone. Species and temperature profiles for converged solutions are depicted on Fig. 2; reaction rates r 1 a n d to 2 are plotted on a log scale in Fig. 3.
Numerical Solution for Gas Phase Equations A eonvenient means of obtaining a solution of the steady-state equations for the gas phase consists of f o r m u l a t i n g p s e u d o - n o n s t e a d y state' equations and solving them by a numerical time-marehing m e t h o d until steady-state is achieved.
REACTION MODEL FOR DEFLAGRATING RDX
In this expression, 2((U *) represents a matrix with coefficients that are functions of U* multiplying the solution vector U ~ The gen" eral implicit algorithm utilized to solve Eq. (18) is:
i
P = 40 AIM 108
1227
p = 20 ATM
(U *'§ - U * ' ) / A t = 0 ' X ( U ~
.~0D
~~o 104
J
~J'l
+ (1 - 0') 2 ( U ' J )
4 6 8 i0 DISTANCEFROMSURFACE,MICRONS
U*'---ZJ
(20)
12
Using a first predicted value of U *j+~
FIG. 2. Species concentrations and temperature profiles, inner flame zone, showing that increased RDX depletion near the surface together with larger gradients are observed as p increases. The conservation equation for species and energy may be written in vector form as:
O0/Ot =
(19)
[ I - O' at2(U*'+')] U *'+' =[I+(1-0')at2(U*')]
2 x,
U*'
Rearranging:
io2 0
U *'+,
L~ ((f)
+
F(0)
(17)
The method used for the numerical solution of these equations is based on approximation of the operators O/Ox and OZ/Ox2 in L 2 + F by central differences resulting in a system of ordinary differential equations:
d U * / d t = X(U*) U*
(18)
where U* is the solution vector expressed in a space of equidistant points across the flame.
U~ J+' = 2 U *~ - U *j-',
(21)
Equation (20) may be written in quasi-linear form as a corrector for U*J+~:
[I-
O' At)((U~J+')] U * ' '
= ZJ
(22)
and is then solved as a linear system by conventional block-tridiagonal Gaussian elimination. The right side of Eq. (22) was computed by the exact predictor: 18 Z j = Z j - 1 -.J- [ U Oj -
ZJ-l]//0
'
(23)
Initial conditions for this pseudo-nonsteady calculation include a temperature profile chosen to insure vigorous reactions and species mass fraction profiles that satisfy the relation ~--1 Y~ = 1 at any point and are physically reasonable.
Liquid Layer Analysis
6o0[ / ~ ~ .
-%~--~ ......
[/,:'.-"
to. ..........
~ 0
2 x,
to.~
7
~
/
......
4 6 8 10 DISTANCEFROMSURFACE,MICRONS
to., 0.1 12
Fro. 3. Reaction rate profiles, inner flame zone, showing increased relative importance of secondary reaction and more distinct inner-outer zone boundary at higher pressure.
Observations that confirm the existence of a liquid layer during steady burning of RDX at low to moderate pressures have been made by several investigators. 19,2~RDX decomposition in the liquid phase for temperatures above the melting point (480 K) has been verified experimentally. 6,s,n Scanning electron micrographs of extinguished samples el after burning at about 20 atm have been made showing entrapped gas bubbles; unfortunately, extinguishment was achieved by rapid depressurization, so one may not be certain about the actual presence of gas bubbles during the deflagration process itself. Detailed models of reacting melt layers have been developed for pyrolytic degradation (endothermic) of polymers z2 and incorporated into a deflagration model of a m m o n i u m perchlorate monopropellant, z3 The advantage
1228
PROPELLANT IGNITION AND COMBUSTION
to detailed reactive melt layer analysis is, in general, that Y1(0-), Y2(0-) ... YN(0-) and d T / d x ( O - ) are solved directly, without need to invoke additional simplifying assumptions (e.g., imposed burning rate data). Further, the analysis may apply to any small (but finite) liquid layer thickness. These may not be obtained from collapsed (zero thickness) reacting sub-surface approximate models that involve overall heat balance. In the remainder of this section, a brief description of the reacting liquid phase model is given. A full description of the development and solutions is given elsewhere. 4 RDX decomposition (to a significant extent) is considered to occur in the melt zone, with the corresponding heat release. This is the only reaction considered in the liquid. Its stoichiometry is given by Eq. (1) and the Arrhenius rate constant is that of Eq. (3). The gaseous products of this reaction are assumed to remain dissolved in the liquid (RDX) and to be convected by the liquid toward the (hot) fixed boundary at x = 0. Molecular diffusion (but not heat diffusion) is assumed to be negligible. The physical properties are assumed to be uniform throughout the liquid zone for the range of temperatures and extent of reaction considered. The conservation equations for the RDX species and energy are, in the region - ~ m+ ~< x_<0-:
dr,
r
A 1 e x p ( - E 1/ R u T )
(24)
dx d2T dx 2
r a
dT dx O~
= A 1e x p ( - E ~ / R , T ) Y ~ A h ~ , q - ~
x~
(25)
The boundary conditions are: r~(-aa) = 1
(26a)
T ( - 8 , . ) = T,.
(26b)
T(O) = T,
(26c)
The liquid layer thickness, 8 m, is not known a priori and has to be calculated. An overall energy balance on the solid phase plus the solid/liquid interface energy balance are utilized to provide additional conditions by which 8,~ is defined. These yield:
The method of solution, in general, involves transforming the given boundary value problem, Eqs. (24), (25), (26), and (27), into the associated initial value problem. Introducing dimensionless temperature and space coordinate: Xl =- ( T -
Ts)/T*
=- x r / e t c
(2Sa) (28b)
and the dimensionless parameters: T ~ ------Ahliq/WiCc, A 1 =---a c A l / r
01 =--- E 1 / B ~ T * , 02 =- T ~ / T * , 2 and a =- g m r / c % . The system
of equations becomes:
dlq dE
= - Y 1 A l e x p [ - 0 1 / ( - q +02)] d~l --= dE
dv
--= dE
v(E)
v(E) + Y, A1 exp[-O~/('q +Oz)]
(29a) (29b) (29c)
The initial conditions are: Y1 ( - a ) = 1 rl(-a) = (T m - Ts)/T* v(-a)
= [Ah m + C c ( T ~ - T o ) ] ~ C o T *
(30a) (SOb) (30c)
The system of Eqs. (29) and (30) may be integrated numerically with little effort. Note that although only YI(E) is calculated, the r e m a i n i n g species concentrations may be found at any point using the stoichiometry of Eq. (1). "q(~), being a continuous, monotonous, single-valued function of ~ in the region - a -< E -< 0-, serves as a measure of the point where the integration is stopped; viz., where Xl = 0(or T = Ts). To obtain a qualitative measure of the solution behavior, a more elementary model was set up. By assuming all heat release to be localized at the x = 0 interface, an analytical solution was obtained using asymptotic methods. The elementary model clearly approximates cases where only a small amount of distributed RDX decomposition occurs in the melt, i.e., a "thin" melt zone. Results of the comprehensive model compared with the elementary model are shown in Figs. 4, 5, and 6.
dT Gas~Melt Interface Conditions
x~-~x ( - ~ ) = m[Ah m + Cc(T m-
To)]
(27)
In the foregoing sections, the gas phase and melt phase processes were treated separately.
REACTION MODEL FOR DEFLAGRATING RDX 465
i
COMPREHENSIVEMODEL APPROXIMATEANALYSIS . . . . . . . . . BOTH CONVERGED(T~,M') SOLUTIONS--(~--
~-,20,0
390
"
i
MASSFLUX
i
i
PRESSURE
?
E
~
1229
315
15.0
0,52 I - - I
z
\\
/ ~ ......
240
\~
165 rMAS~ FLUX PRESSURE I G/CM'LSEC
O, fl / 600
ATM
I I L I 620 640 660 680 Ts, SURFACETEMPERATURE(K)
Fro. 4. Liquid phase layer thickness vs temperature showing decelerated decrease in thickness as regression rate becomes larger. So far, the only parameters they seem to have in c o m m o n are T~ a n d m (for any particular case). Evidently, there is a stronger int e r d e p e n d e n c e b e t w e e n those two elements, in terms of reactant s u p p l y to the gas b y the liquid phase and heat f e e d b a c k from the gas to the liquid. The objective of this section is to describe the l i q u i d / g a s interface conservation conditions and the method e m p l o y e d to incorporate them into the overall deflagration model. The following energy and species conservation conditions s h o u l d be satisfied at the liqu i d / g a s interface, x = 0:
MASSFLUX PRESSURE G/CN2-SEC ATM
" ' - , , . - " - , , , ~ . ~z'~,,,.(40,
.....
921
700
S. . . . . .
620 640 660 680 Ts, SURFACETEMPERATURE(K)
700
FIG. 6. Liquid phase thermal gradients vs temperature showing accelerated increase in dT/dx (0-) as regression rate increases.
dT
dT
% ~-(o+)= xo ~-(o-) + m , ~ h ~ Y, ( 0 -
Y, (0 + )
) (31)
(oD) s dY, - (0 + ) = Y, ( 0 - ) ,
m
dx
i = 1, 2 . . . . N
(32)
In addition, a C l a u s i u s - C l a p e y r o n evaporation law is utilized:
Pvl = Pv~ e x p
- -~u
T~
T O.
(33)
T O and P~ 1 are the reference temperature and the c o r r e s p o n d i n g reference partial pressure of RDX, taken at the melting point. Equations (31) to (33) constitute s system of N + 2 algebraic constraints, involving the following unknowns: Y~(0-),
Y~(0+),
(pD)~dY~/dx(O+), for
i= 1,2,...N
and 0,7
Xg~ dT/dx(O+), '~
0,6%
do
640
d0
dT/dx(O-), T~ and
m
(10)\"x~
680
Ts, SURFACETEMPERATURE(K)
FIG. 5. Liquid phase RDX mass friction vs temperature showing that higher T, of converged solutions overbalances smaller residence time to obtain larger RDX depletion at higher regression rates.
The formulation contains 3 N + 4 unknowns. This n u m b e r is r e d u c e d b y the following considerations: a. Given T~ and m, the solution of the l i q u i d
phase provides Yi(O-), i = 1, 2 . . . . and d r / dx(O- ).
N
PROPELLANT IGNITION AND COMBUSTION
1230
b. Given T,, m and Y~(0+), i = 1, 2 . . . . N [plus the boundary conditions (Eq. 16.b) at x = + % naturally] the solution of the gas phase yields k g, d T / d x (0 + ) and (oD)~ dY, / dx (0 + ), i = 1, 2 . . . . N. Therefore, a system of N + 2 equations with N + 2 unknowns remains, thus resolving the problem of closure. The unknowns are considered in the present framework as the set of independent parameters. Explicitly, their vector is defined as: 12r= (Y~(0+)... YN(0 +) T , m )
(34)
Equations (31)-(33) may now be written in short notation as the system: fj(~)=0,
j = 1,2 .... N + 2
(35)
Note that (in the nontrivial case) fi(0-*) = 0 for all j, only if all the interface conditions are satisfied, i.e., the value of ~* corresponds to a steady-state solution of the deflagration problem at the given ambient conditions. This, of course, is based upon the notion that such a solution is unique, if its exists. In practice, the following iterative procedure is suggested in order to arrive at the desired solution vector, ft*. The process begins by initially approximating ~0, for which gas and liquid phase solutions are generated; then, an attempt to satisfy Eq. (35) is made, resulting generally in: fj(~0)__ejo#0 '
j=l,
2,...N+2
(36)
The residuals or errors ~ 0 indicate that 120 ~ ~*, i.e., that 12~ is n ~ the ~ solution. The next trial vector ~ is obtained by using the Newton-Raphson iterative method; the process is repeated until sufficient convergence, (i.e., g --~ 0) is achieved. To simplify this rather complex general case for practical computations, only 3 out of the total of N + 2 = 11 constraints were actively iterated upon; these were chosen as the most sensitive constraints. Correspondingly, the dimension of ~ was also reduced to 3, taking the most deeply e m b e d d e d parameters; the rest were made dependent upon these 3. The 3 conditions chosen were the energy conservation, RDX species conservation and the evaporation law for RDX. (The remaining constraints were only observed.) The 3 independent parameters chosen were 1:1(0 + ), T~ and m. The Jacobian matrix for the iterative procedure is now defined
J -
(37)
It is formed numerically by repeatedly perturbing the vector 0- by 8~/k) and obtaining the corresponding 8~ Ik), then used in finite difference approximations of 0k j / 0 ~ . Corrections to the (n + 1) iteration-loop Ix are given by 8~.<"+l> = - J - ~ ~ <">
(38)
where (n) denotes the current iteration loop. The t2rocess is repeated until convergence, i.e., --* 0 is obtained. The procedure, being convergent generally, was found to diverge at the vicinity of minimal ]gl. This is due to ~ not being a single-valued function of ~ (although ~ is single valued in ~,). The transformation of Eq. (38) in the vicinity of minimal I~1, is thus not unique without J being a singular transform. In order to overcome this difficulty, the process of finding/2* at minimal 1~1was aided by marching in ~ in order to obtain the corresponding ~. In the region of minimal j~], e 3 that corresponds to the evaporation law, was at least 2 orders of magnitude smaller than % and %. Thus, marching by T, and m (Y1 (0 +) being generated by the evaporation law) proved sufficient for obtaining a map of {(g). Converged solutions are therefore defined by [e,(r**; m*)l = I%(Ts*; m*)l = minimum. Discussion of Results
Converged steady-state solutions tend to have mass fluxes reasonably close to those observed experimentally at the pressures considered. As expected, the margin of uncertainty in both T, and m, as well as the minimal ]El attainable, increase with pressure. Maximal components of 4 are always those corresponding to energy and RDX conservation constraints, Eqs. (31) and (32); at the points of convergence, these amount to 0.04, 0.10, and 0.16 at p = 10, 20, and 40 atm, respectively. Gas phase solutions at p = 10, 20, and 40 atm are plotted in Figs. 2 and 3. For purposes of demonstration the corresponding experimentally observed m were used; these values of m are always within the region of minimal I~l in ~-space, found in this study. Figure 2 shows mass fractions and temperature profiles in the inner flame zone. NO, CO 2, CO and H 2 0 were omitted here because of their small concentrations. It may be observed that gradients become steeper and xf becomes smaller
REACTION MODEL FOR DEFLAGRATING RDX as p increases. This is even more evident in Fig. 3, where o)1 and o)2 profiles are plotted on a log scale. Figures 2 and 3 clearly show that the division of the gaseous flame field into 2 zones becomes more distinct as p increases. To verify the importance of including the secondary reaction, an artificial suppression (by a factor of 0.01) of the C H z O + NO 2 reaction rate resulted in reduced heat feedback to the condensed phase by 7%, 11%, and 16% at p = 10, 20, and 40 atm, respectively, which confirms that the secondary reaction is important in the inner flame zone. Liquid phase solutions are plotted in Figs. 4, 5, and 6. In addition, lines connecting converged (T s, m) solutions were overplotted. Figure 4 shows that the layer thickness and residence time (pc?3m/rrt) decrease with increasing p. Figure 5 indicates that the fraction of RDX reacted in the liquid increases with increasing p; this is a result of two opposing trends: the shorter residence time at higher p decreases RDX decomposition, while higher T~ at higher p increases RDX decomposition. Thus it is seen that the higher T~ corresponding to large p overbalances in favor of more RDX decomposition at the higher pressures. Finally, Fig. 6 shows that XcdTfdx increases in an accelerated manner as p increases.
lated, these can be tested readily within the framework of the present model.
Nomenclature As Ai,/~ C D E~ F f, -~h i Ahti q Ah .... I J k
L 2
M
Conclusions m
A model for pure RDX deflagration was formulated and tested. It describes the processes involved in a reacting melt layer and the inner flame zone, consisting of two simultaneous reactions. A set of calculations were carried out using published kinetic and thermochemical data and Arrhenius-type chemical rates. The trends are in good agreement with the published deflagration data. The plausibility of distinct flame zones at intermediate pressures was demonstrated by solving the comprehensive mathematical model. For completeness, the outer flame zone processes should also be formulated and the solution incorporated by matching the outer and inner flame regions at the x = xf interface. In view of the inner flame zone model solution, this may be done with relative ease. Clearly, the model is most sensitive to the thermochemical and kinetic data, which are not known with great certainity and are not expected to be accurate over the full range of pressures and temperatures. However, as more complete data are obtained and more sophisticated chemical mechanisms are postu-
1231
N P R~ r T U U* V v W~, W x .~ y~
pre-exponential factor; for first order reaction, s e e - ' ; for second order reaction, cm3/grnol-sec coefficients in empirical (linear) equation for specific heat, Co; cal/g-K and cal/g-K 2, respectively heat capacity, cal/g-K diffusion coefficient, c m 2 / s e c activation energy, c a l / m o l source term vector in numerical scheme liquid/gas interface constraint specific heat of formation, of species i, c a l / g specific heat of reaction in liquid, cal/g specific heats of fusion, of vaporization respectively, c a l / g the identity matrix Jacobian matrix, Eq. (37) reaction rate constant; for first order reaction, sec-1; for second order, cm 3/ gmol-sec second order differential operator, xspace total number of simultaneous reactions mass burning rate or mass flux, g /
cm2-sec total number of species pressure, atm. (Note: in equation of state--in c.g.s, units.) universal gas constant, 1.986 cal/ gmo]-K. (Note: in equation of state--in equivalent c.g.s, units). linear regression rate, cm/sec temperature, K solution vector, gas phase. numerical (finite difference) approximation of the solution vector U velocity of gas, c m / s e c dimensionless temperature gradient, Eq. (29b) molecular weight of species i and average molecular weight of mixture respectively, g/tool distance, cm finite difference space operator mass fraction
Greek S~tmbols ct
thermal diffusivity, c m 2 / s e c
1232 ~rn
At e, Ej
PROPELLANT IGNITION AND COMBUSTION melt layer thickness, cm time i n c r e m e n t in quasi-nonsteady numerical scheme vector of residuals and c o m p o n e n t thereof, respectively, in g a s / l i q u i d interface constraints, Eq. (36) rx/e~r d i m e n s i o n l e s s space coordinate in l i q u i d layer (T Ts)/(Ahli q / Cc) , d i m e n s i o n l e s s t e m p e r a t u r e in l i q u i d layer implicit parameter in quasi-nonsteady numerical scheme, 0 -< 0' -< - -
0'
1
k P., P.j
P 1) ik ~k
thermal conductivity, c a l / c m - s e c - K vector of i n d e p e n d e n t parameters, Eq. (34) and c o m p o n e n t thereof, respectively. density, g / c m a stoiehiornetric coefficient of i th species in the k th reaction k th overall reaction rate, t o o l / c m 3 - s e c
Subscripts c
e,f g i j
k p s 0 1
c o n d e n s e d phase denote ends of outer and inner flame zones, respectively gas phase i th species jth conservation constraint at g a s / l i q u i d interface, Eq. (35) k th overall reaction constant pressure g a s / l i q u i d surface a m b i e n t conditions; also, p r e d i c t e d solutions in numerical scheme denote RDX species, i.e., i = 1
"2. BATTEN,T. T.: Australian J. Chem. 23, 737 (1970). Also, Aus. J. Chem. 23, 749 (1970) and Dept. of Supply, Australian Defense Scientific Service, Defense Standards Laboratories, Rept. No. 412 (1971). 3. CoscRovE, J. D. AND OWEN, A. J.: Combustion and Flame, 22, 13 (1974) and also Combustion and Flame, 22, 19 (1974). 4. BENREUVEN,M.: Ph.D. Dissertation in preparation, Princeton University, Princeton, N.J. 5. FLAXAGA.~,D. A,: Personal communication, January 1976. 6. HALL,P. G.: Trans. Farad. Soc., 67 (Pt. 2), 556 (1971). 7. ROSEN, J. M, AND DICKINSON,D.: J. Chem. Eng. Data, 14, 120 (1969). 8. ROBERTSON,A. J. B., Trans. Farad. Sot,, 45, 85 (1949). 9. SMITH,B. S.: "DTA and DTGA Analysis Studies of RDX and HMX," NWL Technical Report, TR-2316, July 1969, U.S. Naval Weapons Laboratory, Dahlgren, VA. 10. ROGERS,R. N. ANDSMITH,L. C.: Thermochemica Acta, 1970, pp. 1-9. 11. ROGERS, R. N. AXD DArn, G. W.: Anal. Chem. 45, 3, 1973, p. 596. 12, POLLARD,F. H. AND WYATT, R. M. H.: Trans. Farad. Soc., 45, 760 (1949). 13. FWER, R.: Personal Communication, PittmanDun Laboratory, Frankford Arsenal, Philadelphia, Pa., Nov. 1075. 14. WILLIAMS, F. A.: Combustion Theortj--The
Fundamental Theo~ of Chemically Reacting Flow Systems, pp. 1-17, 390-403, Addison15.
Superscripts 0
(k) (n) J T
initial a p p r o x i m a t i o n in g a s / m e h interface constraints, iteration scheme p e r t u r b a t i o n n u m b e r in iteration scheme iteration number, Eq. (38) n u m b e r of time step in n u m e r i c a l scheme, t = j A t transpose (for vector or matrix)
16. 17. 18.
19. REFERENCES 1. RAUCH,F. C. AND FANELLI,A. J.: J. Phys. Chem, 73, 1604 (1969).
20.
21.
Wesley Publishing Co., Inc., Reading, Mass., 1965. TOULOUKIAN,Y. S., LILLY, P. E. AND SAXENA,S. C.: Thermophysical Properties of Matter, the TPRC Data Series, IFI/Plenum, N.Y.-Wash., 1970. Also: Gallant, R. W., Physical Properties of Hydrocarbons, C1-C4, Aldehydes, Gulf Publishing Co., Houston, Texas, 1968. BIRD, R. B., STEWART,W. E. AND L1GHTFOOT,E. N.: Transport Phenomena, Chapt. 8, pp. 243264, J. Wiley, N.Y., 1960. SHELVA,R. A., McBmDE, B. J.: NASA TND-7056, NASA, Washington, D.C., Jan. 1973. VICHNEVETSk'Y,R.: Report NAM 153, Department of Computer Science, Rutgers University, New Brunswick, N.J., August 1974. TAYLOR,J.: Trans. Faraday Soc., 58, 471, pp. 561-568, 1962. BOBOLEV,V. K., MARGOLIN,A. M. AND CHUIKO, S. V.: ]. of Combustion, Explosion and Shock Waves, Vol. 2, No. 4, pp. 24-32, 1964. DERB,R. L., BOGGS,T. L., ZVRN,D. E. ANDDIBBLE, E. J.: Proceedings of the 11th JANNAF Combus-
REACTION M O D E L FOR D E F L A G R A T I N G RDX
tion Meeting, Vol. 1, pp, 231-242, CPIA Publication #261, Chemical Propulsion Information Agency, Sept. 1974.
1233
22. LAYCEt.LE, G.: AIAA J., 8, 1989-1996, (1970). 23. G~-lP,nO, C. AXD WILLIAMS, F. A.: AIAA J., 9, 1345-1356 (1971).
COMMENTS N. Kubota, Japan Defense Agency, Japan. The deflagration mechanism of RDX was described in terms of two reaction steps expressed by Eqs. (3) and (4), and the reactions between NaO and C H 2 0 were assumed to be sufficiently slow to affect the rate of the deflagration in this paper, However, the secondary gas phase reaction involving NO reaction is expected to be strong exothennic reaction, and the heat feedback from this secondary reaction zone to the first reaction zone possibly accelerates the reaction expressed by Eqs. (3) and (4). Did you examine the effect of this secondary reaction on the deflagration rate of RDX? The gas phase reactions involving NO is considered to be fast enough when the temperature and the pressure are high. The photographic observations of RDX crystal show that the luminous flame of RDX appears very close to the b u r n i n g surface of the crystal and is not like the case of the luminous flames of nitrate ester based propellants. Authors'Reply. The present paper's main concern, in terms of gas phase analysis is the "near-field" flame zone, where gaseous RDX decomposition and NO 2 + C H 2 0 reactions dominate. "Outer" flame field processes, in comparison, may have higher order, (namely smaller) effects upon the overall deflagration rate and the heat feedback to the condensed phase.
In particular, NO + C H 2 0 reaction is quite improbable due to high expected activaton energy, even at temperatures in the neighborhood of 1000~ (Notethat the N~----Ob o n d energy is 149.7 kcal/mole, being quite high at the pressure range considered.) To demonstrate this point with a reaction that is more probable at the outer field, consider the reaction rate of N~O + C H e O at the end of the inner flame zone; using the converged results of a 20 atm run and available data, approximately: A 3 - 1.2 • 1013 c m a / g m o l - s e c
and
E 3 - 44000 c a l / g m o l SO: toz = 72 x ka(1200 ) x (0.21/0.044)(0.42/0.030) = 3.8 x 102 mol/ma-sec while for NO 2 + C H 2 0 at the same point, ~0z 10 v mol/ma-sec. Therefore, the heat produced b y the N 2 0 + C H 2 0 reaction may not affect the thermal gradient (at the point considered) appreciably--at least not to first order. Finally, it should be noted that the t h i n luminous zone adjacent to the condensed surface is not observed for pure nitramine crystals deflagrating at 21 atm.